Effective algorithms for solving functional equations with superposition on the example of the Feigenbaum equation

Abstract

Purpose. New algorithms were consider for functional equations solving using the Feigenbaum equation as an example. This equation is of great interest in the theory of deterministic chaos and is a good illustrative example in the class of functional equations with superposition. Methods. The article proposes three new effective methods for solving functional equations — the method of successive approximations, the method of successive approximations using the fast Fourier transform and the numerical-analytical method using a small parameter. Results. Three new methods for solving functional equations were presented, considered on the example of the Feigenbaum equation. For each of them, the features of their application were investigated, as well as the complexity of the resulting algorithms was estimated. The methods previously used by researchers to solve functional equations are compared with those described in this article. In the description of the latter, the numerical-analytical method, several coefficients of expansions of the universal Feigenbaum constants were written out. Conclusion. The obtained algorithms, based on simple iteration methods, allow solving functional equations with superposition without the need to reverse the Jacobi matrix. This feature greatly simplifies the use of computer memory and gives a gain in the operating time of the algorithms in question, compared with previously used ones. Also, the latter, numerically-analytical method made it possible to obtain sequentially the coefficients of expansions of the universal Feigenbaum constants, which in fact can be an analytical representation of these constants.